Problem 38 - Contemporary Precalculus

Question

In Exercises \(37-52,\) express the number in polar form. $$5-5 i$$

Step-by-Step Solution

Answer

Answer: The polar form of the complex number $$5-5i$$ is $$5\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))$$.

Step 1: Find the modulus (magnitude) of the complex number

To find the modulus of the given complex number, we can use the formula $$r = \sqrt{a^2 + b^2}$$, where $$a$$ is the real part and $$b$$ is the imaginary part. In our case, $$a = 5$$ and $$b = -5$$, so: $$r = \sqrt{(5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}$$

Step 2: Find the argument (angle) of the complex number

To find the argument of the complex number, we can use the formula $$\theta = \arctan(\frac{b}{a})$$ (keeping in mind that $$\theta$$ should be in the correct quadrant). In our case, $$a = 5$$ and $$b = -5$$, so: $$\theta = \arctan(\frac{-5}{5}) = \arctan(-1)$$ Since both the real part and the imaginary part are positive, the complex number lies in the fourth quadrant. Therefore, the angle should be negative, which corresponds with the arctan calculation above. The angle for $$\arctan(-1)$$ is $$-45°$$ or $$-(\frac{\pi}{4})$$ radians.

Step 3: Write the polar form

We now have the modulus of the complex number, $$5\sqrt{2}$$, and its argument, $$-\frac{\pi}{4}$$ radians (or $$-45°$$). To write the given complex number in polar form, we use the formula $$r(\cos(\theta) + i\sin(\theta))$$ with the values we found: $$5\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))$$ So, the polar form of the complex number $$5-5i$$ is $$5\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))$$.

Key Concepts

Complex Numbers

Complex numbers are an extension of real numbers and are essential in mathematics for describing phenomena that real numbers cannot. A complex number is generally represented as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, with \(i\) being the imaginary unit, defined by the property \(i^2 = -1\). Any number of the form \(a + 0i\) is a real number, and \(0 + bi\) constitutes an imaginary number.
Complex numbers have a two-dimensional plane representation known as the complex plane, where the x-axis represents the real part, and the y-axis represents the imaginary part. This visual allows for enhanced analysis of operations involving complex numbers, providing insights into both their magnitude and angle.

Modulus of a Complex Number

The modulus of a complex number, denoted as \(r\), is akin to the "length" of the vector represented by the complex number in the complex plane. It is the distance from the origin to the point \((a, b)\). To find the modulus, use the formula:

• \( r = \sqrt{a^2 + b^2} \)

This formula is derived from the Pythagorean theorem and gives insight into the magnitude of the complex number. For example, for the complex number \(5 - 5i\), the modulus is calculated as:
\[r = \sqrt{(5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}.\]This modulus indicates that the complex number is \(5\sqrt{2}\) units away from the origin.

Argument of a Complex Number

The argument of a complex number is the angle formed between the positive x-axis and the line representing the complex number in the complex plane. Denoted as \(\theta\), it provides information on the direction of the complex number. Calculate the argument using:

• \( \theta = \arctan\left(\frac{b}{a}\right) \)

When determining \(\theta\), it is vital to consider the quadrant in which the complex number lies. This is because the arctangent function alone provides angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), thus requiring adjustment based on the signs of \(a\) and \(b\). For the complex number \(5 - 5i\):
\[\theta = \arctan\left(\frac{-5}{5}\right) = \arctan(-1) = -\frac{\pi}{4}\]This angle reflects the complex number's position in the fourth quadrant, where both real and imaginary components are positive based on the complex plane’s orientation.

Conversion Between Rectangular and Polar Coordinates

Converting complex numbers between rectangular (standard form) and polar coordinates provides flexibility in problem-solving. In rectangular form, a complex number is \((a, b)\), but in polar form, it is expressed as \(r(\cos\theta + i\sin\theta)\), where \(r\) is the modulus, and \(\theta\) is the argument.

• To convert from rectangular to polar:
  • Calculate modulus \(r\) using \(r = \sqrt{a^2 + b^2}\).
  • Determine argument \(\theta\) using \(\theta = \arctan\left(\frac{b}{a}\right)\), considering the correct quadrant.
  • Write the polar form as \(r(\cos\theta + i\sin\theta)\).
• To convert back to rectangular:
  • Use the relationships \(a = r\cos\theta\) and \(b = r\sin\theta\).

For the complex number \(5 - 5i\):

• The modulus is \(5\sqrt{2}\).
• The argument is \(-\frac{\pi}{4}\).
• Hence in polar form: \(5\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))\).